Making sense of Math in Vis Gordon Kindlmann University of Chicago - PowerPoint PPT Presentation

Making sense of Math in Vis Gordon Kindlmann University of Chicago glk@uchicago.edu

(from seminar description) http://www.dagstuhl.de/en/program/calendar/semhp/?semnr=18041 Mathematical foundations of visual data analysis . There is a rich tradition of mathematical and computational methods used in visualization , such as topological approaches, feature extraction, numerical sampling and reconstruction methods, numerical integration, di ff erential operators, fi ltering, dimension reduction, and applications of information theory, partly incorporating uncertainty. While all these methods have a solid mathematical foundation, a careful look at the relation between theories and their role in visual data analysis is needed. How can it all be organized? What’s missing?

Schematic view of Visualization pipeline interaction loops Visualization compute hardware

Why have Math in Vis? To Describe , to Abstract away speci fi cs, embrace a level of generality ⇒ Usefully Structure Visualization Pedagogy To Connect , to Engage with disciplines that already use math To Enrich/Solidify vis, by leveraging formalism of mathematics To Aspire to or Broadcast Sophistication?

Where is Math in Vis? In about 6 di ff erent Help! 
 places correct (now) 
 & 
 complete (this week)

(1) in models of (scienti fi c computing) objects/phenomena Laplace’s equation, of interest Poisson’s equation Navier-Stokes, Heat, Advection-Di ff usion PDEs Re fl ection, Illumination, Energy Transport Statistics: Ensembles, Uncertainty, Bayesian Methods

(2) in models of human (Perceptual Psychology) perception and Stevens Law, Weber- cognition Fechner Law Opponent Color Channels, Color Appearance Models (e.g. CIECAM02) Gabor Wavelets (perception of scale) Bayesian Models Of Gestalt [Jäkel-QuantitativeGestalt-VR-2016]

(3) in empirical study of how people perform some task using vis (Statistics of Experimental Psychology) Randomization, Counterbalancing Signi fi cance levels, P-values, ANOVA [Forsell-IntroEvalVis-HHCV-2014]

(4) in de fi nitions of essential overall goal of vis method Feature Extraction : Isocontours, Parallel Vectors Operator, Ridges and Valleys, Vortex Cores Topological methods : Morse-Smale Complex, Reeb Graph Points & Graphs : Principal Component Analysis, Spectral Clustering, Dimensionality Reduction, Graph Drawing Solving PDEs with Finite Elements : Dirichlet/Neumann boundary conditions, Galerkin Method, Linear/Spectral Elements A B

(5) to implement low-level parts of overall method Linear algebra : LU Computing w/ decomposition, eigensolve Approximations : Taylor Series, Fourier Series, Numeric Methods : Euler/ Wavelets Runge-Kutta Integration, Streamlines, Tractography, Signal processing : Nyquist Newton root fi nding, Sampling, Reconstruction by Newton optimization, Convolution Kahan summation Sequence Data : matching, Derivatives : Gradient, searching, Smith–Waterman Jacobian, Laplacian, alignment Hessian


 (6) to characterize performance of compute hardware for task IEEE754 fl oat point Measures of: speedup, scalability, locality, granularity, bandwidth

What’s missing? Math in Math in inner loop outer loop of vis of vis

Math of vis (not in vis) Mathematical description of what it means to be a visualization

William Hibbard [Hibbard-StructuresOfData-DIDV-1995] [Demiralp-VisualEmbedding-CGnA-2014]

Dagstuhl Seminar λ 2 Worksheet for design of 2D tensor glyphs Gordon L. Kindlmann 09251 Scienti fi c (+ , +) Visualization Worksheet for design of 2D λ 1 symmetric tensor glyphs (July 2009) (+ , − ) “Symmetry” ( − , − ) “Continuity” [Schultz-Superquad2-VIS-2010] (+,+,-) (+,+,+) (+,-,-) (-,-,-)

Mathematically abstracting vis design T Transforms 
 T’ Transform 
 glyph 
 Data Vis T, T’ V(X), V(T(X)) property V(X) X Re fl ection, V “Symmetry” Rotation T’ T “Continuity” V Perturbation T(X) T’(V(X)) ≈ [Kindlmann-AlgebraicVisDesign-VIS-2014] T ≈ T’ ⇒ V(T(X)) Concrete Terminology for Problems: Hallucinators, Confusers, Jumblers, Misleaders [Chen-InfoTheory-TVCG-2010]

Correspondence example: elevation colormap Data: signed diverging elevation relative to colormap sea level v D α (e) = -e ω : negate hue –v(e) ≈ v(–e) v D colormapping commutes with negation

Correspondence example: elevation colormap Data: signed ω = negate hue elevation relative to sea level v D α (e) = -e ?? v D meaningful α not matched with perception: “jumbler”

(from seminar description) http://www.dagstuhl.de/en/program/calendar/semhp/?semnr=18041 Mathematical foundations of visual data analysis . There is a rich tradition of mathematical and computational methods used in visualization , such as topological approaches, feature extraction, numerical sampling and reconstruction methods, numerical integration, di ff erential operators, fi ltering, dimension reduction, and applications of information theory, partly incorporating uncertainty. While all these methods have a solid mathematical foundation, a and algebra careful look at the relation between theories and their role in visual data analysis is needed. Anything else missing? Better way to organize?

Conclusions Short of unifying “theory of vis”, need accounting of math in vis (please help) Why: principles empower vis students (more than a craft taught by apprenticeship) Math of vis essential for Theory of vis ⇒ Either way, Vis needs Math

References [Chen-InfoTheory-TVCG-2010] An information-theoretic framework for visualization . M Chen and H Jänicke. IEEE Transactions on Visualization and Computer Graphics, 16(6):1206–1215, 2010. [Forsell-IntroEvalVis-HHCV-2014] An Introduction and Guide to Evaluation of Visualization Techniques Through User Studies . C Forsell and M Cooper. In Handbook of Human Centric Visualization, pages 285–313. Springer New York, New York, NY, 2014. [Hibbard-StructuresOfData-DIDV-1995] Mathematical Structures of Data and Their Implications for Visualization . WL Hibbard. In Proceedings of the IEEE Visualization '95 Workshop on Database Issues for Data Visualization, pages 76–85, London, UK, UK, 1996. Springer-Verlag. [Jäkel-QuantitativeGestalt-VR-2016] An overview of quantitative approaches in Gestalt perception . F Jäkel, M Singh, FA Wichmann, and MH Herzog. Vision Research, 126:3 – 8, 2016. Quantitative Approaches in Gestalt Perception. [Kindlmann-AlgebraicVisDesign-VIS-2014] An Algebraic Process for Visualization Design. G Kindlmann and C Scheidegger. IEEE Transactions on Visualization and Computer Graphics (Proceedings VIS 2014), 20(12):2181–2190, November 2014. [Schultz-Superquad2-VIS-2010] Superquadric Glyphs for Symmetric Second-Order Tensors . T Schultz and GL Kindlmann. IEEE Transactions on Visualization and Computer Graphics (Proceedings of VisWeek 2010), 16(6):1595–1604, November–December 2010. http://www.oscarmayer.com/